APL Pho onics ARTICLE sci a ion.o g/jou nal/app
Single-sho supe - esolu ion quan i a i e phase
imaging allowed by cohe ence ga e shaping
Ci e as: APL Pho on. 8, 046103 (2023); doi: 10.1063/5.0127950
Submi ed: 25 Sep embe 2022 •Accep ed: 10 Ma ch 2023 •
Published Online: 3 Ap il 2023
Mi osla ˇ
Du iˇ
s,1,a) Pe Bouchal,1,2 and Radim Chmelík1,2
AFFILIATIONS
1CEITEC–Cen al Eu opean Ins i u e o Technology, B no Uni e si y o Technology, Pu kyˇ
no a 656/123, 61200 B no, Czech
Republic
2Ins i u e o Physical Enginee ing, Facul y o Mechanical Enginee ing, B no Uni e si y o Technology, Technická 2896/2, 61669
B no, Czech Republic
a)Au ho o whom co espondence should be add essed: [email p o ec ed]
ABSTRACT
Biomedical and me asu ace esea che s epea edly each o quan i a i e phase imaging (QPI) as hei p ima y imaging echnique due o i s
high- h oughpu ,label- ee,quan i a i ena u e.So a , e yli lep og esshasbeenmade owa dachie ingsupe - esolu ioninQPI.Howe e ,
hepossiblesupe - esol ingQPIwouldsa is y heneed o quan i a i eobse a iono p e iouslyun esol edbiologicalspecimen ea u esand
allow unp eceden ed h oughpu s in he imaging o dielec ic me asu aces. He e we p esen a me hod capable o eal- ime supe - esolu ion
QPI, which we achie e by shaping he cohe ence ga e in he holog aphic mic oscope wi h pa ially cohe en illumina ion. Ou app oach is
based on he ac ha he poin sp ead unc ion (PSF) o such a sys em is a p oduc o he di ac ion-limi ed spo and he cohe ence-ga ing
unc ion, which is shaped simila ly o he supe oscilla o y ho spo . The p oduc simul aneously p oduces he PSF wi h a supe - esolu ion
cen alpeakandminimizessidelobee ec scommonlyde alua ing hesupe oscilla o yimaging.Theminimiza iono sidelobesand esolu ion
imp o emen co-occu in he en i e ield o iew. The e o e, o he i s ime, we achie e a single-sho wide ield supe - esolu ion QPI. We
demons a ehe e esolu ionimp o emen onsimula edas wellas expe imen alda a.Aphase esolu ion a ge image showsa esol ingpowe
imp o emen o 19%. Finally, we show he p ac ical easibili y by applying he p oposed me hod o he imaging o biological specimens.
©2023 Au ho (s). All a icle con en , excep whe e o he wise no ed, is licensed unde a C ea i e Commons A ibu ion (CC BY) license
(h p://c ea i ecommons.o g/licenses/by/4.0/). h ps://doi.o g/10.1063/5.0127950
I. INTRODUCTION
Fa - ield luo escen supe - esolu ion echniques such as s im-
ula ed emission deple ion,1s uc u ed illumina ion mic oscopy,2
pho oac i a ed localiza ion mic oscopy,3and s ochas ic op ical
econs uc ion mic oscopy4ha e become, o e ecen yea s, a s an-
da d in biomedical imaging. These me hods p oduce images wi h
spa ial esolu ion eaching alues way below he di ac ion limi
o ligh . The echniques men ioned abo e exploi sub-di ac ion
limi edimaging o non-linea specimen esponsesachie ed by label-
ing wi h luo escen dyes o quan um do s. A i icial labeling is
also popula o p o iding a high deg ee o speci ici y. Howe e ,
se e al s udies ha e shown ha labeling changes he beha io o
he s udied biological specimen.5,6 The e o e, label- ee imaging
echniques a e a mo e app op ia e choice in many biomedical appli-
ca ions. No need o labeling also allows o s udying a i icial mic o
and nanos uc u es.7,8 None heless, b eaking he di ac ion limi
in label- ee imaging echniques is mo e challenging because o he
missing non-linea specimen esponse.9
Quan i a i e phase imaging (QPI) has es ablished an i e-
placeable ole among label- ee imaging echniques hanks o i s
capabili y o quan i a i ely measu e mo phology and in insic spec-
imen con as wi h nanoscale sensi i i y.10 The possible supe -
esolu ion QPI will sa is y he need o quan i a i e obse a ion
o p e iously un esol ed specimen ea u es and allow inc easing
he space-bandwid h p oduc (SBP),11 c ucial o high- h oughpu
s udies. High SBP is impo an in iden i ying a e e en s, o
example, in d ug disco e y,12 cance -cell biology,13,14 o s em-cell
APL Pho on. 8, 046103 (2023); doi: 10.1063/5.0127950 8, 046103-1
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APL Pho onics ARTICLE sci a ion.o g/jou nal/app
esea ch.15 The ecen de elopmen o au oma ed da a analysis
and classi ica ion by a i icial in elligence16,17 exagge a es his e e -
inc easing demand o high- esolu ion quan i a i e da a. So a ,
he p oposed app oaches o QPI supe - esolu ion a e based on
oblique illumina ion,18,19 s uc u ed illumina ion,20 and speckle
illumina ion,21,22 which, combined wi h pos -p ocessing, p o ide
syn he ic images wi h an e ec i ely enla ged nume ical ape u e
(NA).Thesesyn he icape u eme hodsenhance esol ingpowe by
essen ially mul iplexing he spa ial- equency con en o he objec
spec um in o an unused deg ee o eedom in he sys em, sac i ic-
ing acquisi ion speed, quan i a i e in o ma ion accu acy, o a ield
o iew (FOV).
Recen ad ances in supe oscilla o y ho spo c ea ion23–25 ha
allowed he de elopmen o no el app oaches o cohe en label-
ee supe - esolu ion mic oscopy could also be adop ed o QPI.
Howe e ,cu en implemen a ionso supe oscilla ionsalso sac i ice
some o he aluable mic oscope p ope ies simila o he syn he ic
ape u e me hods. Band-limi ed ields con aining supe oscilla ions
oscilla e locally as e han he highes Fou ie componen . When
ca ied o e o op ical imaging, his means ha he ocal spo can
be made much smalle han allowed by he Abbe–Rayleigh limi .
This was i s in es iga ed in 1952 by di F ancia,26 bu only ecen ly
ha e hese p inciples been applied o p ac ical mic oscopy.27,28
A supe oscilla o y sub-di ac ion limi ed ocal ho spo can be
p oduced, o example, by cohe en ly illumina ing a specially
designed mask o concen ic annuli o a ying complex ansmis-
sion and wid hs.27 The concen ic annuli mask design can push he
cen al ho spo adius signi ican ly beyond he di ac ion limi , bu
a he cos o high-in ensi y sidelobes,27 which deg ade he image
quali y in s anda d wide- ield imaging. An al e na i e app oach o
ampli ude and phase modula ion is he applica ion o ligh s a es
wi h spa ially s uc u ed pola iza ion, such as he ocusing o adi-
ally and azimu hally pola ized Lague e–Gaussian beams.29,30 The
pionee ing expe imen al esea ch u ilizing supe oscilla ions ini ially
demons a ed he supe - esolu ion imaging only in a e y small
FOV27 dic a ed by he dis ance o he i s high-in ensi y sidelobe.
To emo e he FOV cons ain , Roge s e al.28 combined con o-
cal de ec ion wi h supe oscilla o y illumina ion. They c ea e he
supe - esolu ion image hanks o he cohe en illumina ion pa e n
wi h a sub-di ac ion limi ed cen al ho spo and s ong sidelobes.
Subsequen ly, con ocal de ec ion elimina es he image dis o ing
sidelobe e ec s a he cos o scanning he illumina ion pa e n.
Despi e he g ea po en ial o esolu ion imp o emen , in en-
si y imaging does no apply o mos biological and o he weakly
sca e ingspecimens and lacksquan i a i e in o ma ion. Implemen-
a ion o simila p inciples in QPI is hus a desi able ye challenging
ask due o he complexi y and suscep ibili y o in e e ome ic
sys ems.
In his pape , we p opose a me hod ha does no ha e o
sac i iceanyo he a o ablemic oscopep ope ies oachie esupe -
esol ed QPI. To he bes o ou knowledge, we show o he
i s ime ha pa ially cohe en b oad-sou ce in e e ome e s a e
capableo single-sho wide ieldsupe - esolu ionimagingbyshaping
he so-called cohe ence ga e.31 Ou app oach is based on he ac
ha he poin sp ead unc ion (PSF) o he pa ially cohe en sys-
em is a p oduc o he shaped cohe ence-ga ing unc ion19 (CGF)
and he unc ion desc ibing he di ac ion-limi ed image spo (Ai y
pa e n). We shape he CGF by manipula ing he illumina ion
in he conjuga ed sou ce plane simila ly o he supe oscilla o y
ho spo c ea ion echniques. The p oduc o he supe oscilla o y
CGF wi h he Ai y spo c ea ed by he objec i e in he objec a m
minimizes he sidelobes in he unbounded egion while he CGF
cen al peak deli e s he supe - esol ing powe . The minimiza ion
o sidelobes and esolu ion imp o emen co-occu in he en i e
ield o iew and allow single-sho wide ield imaging. The imaging
hus esembles con ocal de ec ion bu wi h pa allel il a ion o all
image poin s in he ield o iew. The images main ain quan i a-
i e phase in o ma ion and ex end he po en ial o supe oscilla ions
owa d he QPI.
We i s demons a e he e ec s o he supe oscilla o y CGF
using simula ed da a. Then, due o he highly abe a ed pupil plane
o ou expe imen al se up, we ocus in he expe imen al pa on
a limi ing case be ween he supe oscilla o y and supe - esolu ion
CGF. In bo h si ua ions, he ho spo wid h is below he Rayleigh
c i e ion. The dis inc ion c i e ion be ween he supe - esolu ion
unc ionand hesupe oscilla o yonewasp oposedbyHuange al.24
(we p o ide mo e de ails on he de ini ion o he supe oscilla o y
and supe - esolu ion ocal spo in he supplemen a y ma e ial). We
c ea e he CGF in his limi ing case by using a simple ampli ude
annula mask, which p o es expe imen ally obus . We demon-
s a e expe imen ally QPI esolu ion enhancemen using only he
limi ing case, bu he p inciple o ou me hod is ex endable o
he supe oscilla o y ocal spo egion, p omising highe esolu ion
imp o emen . An expe imen wi h a phase esolu ion a ge shows a
esol ing powe imp o emen o 19%, and we show p ac ical easi-
bili y by applying he p oposed me hod o he imaging o biological
specimens.
II. OPTICAL SETUP DESCRIPTION
The p oposed p inciples gene ally apply o a ious pa ially
cohe en in e e ome ic sys ems. Wi hou loss o gene ali y, we
will u he desc ibe he op ical se up and heo e ical amewo k o
he used cohe ence-con olled holog aphic mic oscope32 (CCHM),
comme cially a ailable as he Teligh Q-Phase. The op ical se up
(see Fig. 1) is an adap a ion o he Mach–Zehnde in e e ome e .
I consis s o an objec and e e ence a m con aining wo op ically
equi alen mic oscope sys ems. This holog aphic se up gua an ees
o -axis holog am o ma ion in he in e e ence plane (IP) o b oad
sou ces o an a bi a y deg ee o cohe ence. The possibili y o using
FIG. 1. Op ical se up o he cohe ence-con olled holog aphic mic oscope: S,
ligh sou ce; IF, in e e ence il e ; L, elay lens; BS, beam spli e s; M, mi o s;
Mm, mo able mi o s; C, condense s; O, objec i e lenses; TL, ube lenses; DG,
di ac ion g a ing; OL, ou pu lenses; IP, in e e ence plane.
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APL Pho onics ARTICLE sci a ion.o g/jou nal/app
pa ially cohe en sou ces is p o ided by he di ac ion g a ing
(DG; ansmission phase g a ing wi h g oo e equency 150 mm−1,
blazed a 760 nm o he i s di ac ion o de ) implemen ed in he
e e ence a m acco ding o p inciples p oposed by Lei h and Upa -
nieks.33 In ou sys em, an LED (LED Engin LZ4-00R208, peak
wa eleng h a 660 nm, powe up o 2.9 W) is used o illumina ion
o p o ide a spa ially b oad incohe en sou ce, and he illumina ing
ligh ismadequasi-monoch oma ica e passing hein e e ence il-
e (IF) wi h a cen al wa eleng h o 660 and 10 nm ull wid h a hal
maximum. The sou ce is imaged by a pai o ach oma ic double s
(simpli ied as L in Fig. 1; ocal leng hs 63.5 and 350 mm) h ough
a beam spli e (BS) o he on ocal planes o he condense s
(C;NikonLWDcondense lenses, 0.52 NA, wi h adjus able ape u e
s op). This plane in objec and e e ence a ms and espec i e con-
dense p ope ies can be desc ibed acco ding o Re . 34 by he pupil
unc ions PCo(K )and PC (K ), espec i ely, whe e K =(Kx,Ky)is
he ans e se wa e ec o o a plane wa e behind condense s. The
coo dina es o K a e p opo ional o he espec i e sou ce poin
(pupil-plane) coo dina es. Fo his eason, pupil p ope ies can be
cha ac e ized by a unc ion o K . We use educed wa e ec o no a-
ion ∣K∣=1/λ, whe e λis he wa eleng h o ligh , and K=(K ,Kz)
=(Kx,Ky,Kz), whe e Kz=√∣K∣2−∣K ∣2. We modula e he con-
dense pupilplanes op oduce he sub-di ac ionlimi edcohe ence
ga e, as explained in Sec. III. The undamen al image p ope ies also
depend on he pa ame e s o he objec and e e ence a m objec i e
lenses (O; Nikon Plan Fluo i e Objec i es, 10x/0.3 NA/16 mm WD)
in combina ion wi h ube lenses (TL; Nikon, ocal leng h 200 mm),
cha ac e ized by he pupil unc ions POo(K )and PO (K ). S eppe
and piezo mo o s p o ide ine adjus men o he mic oscope op ical
componen s, which we use o he measu emen o he cohe ence-
ga ing unc ion. The holog ams a e eco ded in IP using an Ando
Zyla 4.2 sCMOS came a.
As shown in Fig. 1, we place he phase o ampli ude mask
in one o bo h o he on ocal planes o he condense s. We
designed he masks o shape he CGF when imaging wi h 10x/0.3
NA objec i e lenses. In simula ions, we assume he phase mask is
composedo concen ic annuli, wi h hephaseshi being ei he 0 o
π adians. We also ca ied ou simula ions wi h he ampli ude mask
subsequen ly used in expe imen s. The ampli ude mask is a single
annulus cu by a lase cu e in o a me al shee . An inne ci cle o
he annulus has a diame e o 16.4 mm. The ou e ci cle diame e
is abou 18 mm, bu mo e impo an ly, he pupil diame e in he
on ocal plane o he condense s is limi ed by he ape u e s op o
∼17.3 mm (co esponding o 0.30 condense NA).
III. THEORY
Quan i a i e phase in o ma ion can be ex ac ed om he mea-
su ed holog ams. As we wo k wi h he o -axis holog aphic se up,
we econs uc holog ams by ca ie emo al in he Fou ie plane.32
In pa ially cohe en sys ems, he holog am c oss-co ela ion e m
depends on he ans e sal displacemen Δq=(Δx,Δy)and ela-
i e ime-delay τo he objec -sca e ed and e e ence ields. The
c oss-co ela ion unc ion is con enien ly desc ibed by a mu ual
cohe ence unc ion31 (MCF) Γ(q,q−Δq,τ)o he wo ields, whe e
q=(x,y)is heposi iono apoin in heimageplanespeci iedby he
coo dina eso heop icallyconjuga ed poin in heobjec plane. The
modulusand phaseimage o pa icula Δqand τa eob ained as he
modulus and a gumen o Γ, espec i ely. The in e e ome ic imag-
ing o a gi en ime-delay τand ans e se displacemen Δqcan be
called a pa ial MCF measu emen .31 The comple e MCF is acqui ed
bymeasu ingand econs uc ingholog ams o allaccessibleΔqand
τ. In his wo k, we use in expe imen s quasi-monoch oma ic illumi-
na ion. The e o e, he in luence o empo al cohe ence is minimal
and mani es s mainly as a speckle noise educ ion. We se τ=0 a
he beginning o each expe imen . The s anda d imaging condi ions
in low-cohe ence in e e ome e s a e when Δq=(0,0). We use his
se ing o he majo i y o ou expe imen s. Howe e , as we show
u he , he comple e MCF measu emen and hence he manipula-
ion wi h Δqis c ucial o a measu emen o he cohe ence-ga ing
unc ion. Ou u he analysis will s ay wi hin he limi s o scala
wa e app oxima ion. Mo e de ailed ma hema ical de i a ions o he
ollowing equa ions a e p o ided in he supplemen a y ma e ial. I
we assume comple e spa ial sou ce incohe ence, τ=0, and Δqas a
pa ame e , he exp ession o he measu ed MCF, has acco ding o
Re . 19, he o m
Γ(q;Δq)= (q)⊗h(q;Δq),(1)
whe e (q)is a complex ansmission o he specimen, he symbol ⊗
deno es con olu ion, and h(q;Δq)=po(q)G∗(q−Δq)is a PSF o
he imaging sys em, whe e po(q)=∬POo(K )exp(2πiK ⋅q)d2K
and
G(q)=∬P∗
Co(K )PC (K )PO (K )exp(2πiK ⋅q)d2K . (2)
We call unc ion G(q) he cohe ence-ga ing unc ion19,31 (CGF).
The in eg a ion egions in po(q)and G(q)a e gi en by he ex en
o he pupil unc ions POo(K )andP∗
Co(K )PC (K )PO (K ), espec-
i ely. These bounda ies de ine he band-limi o po(q)and G(q).
The CGF p o ides il e ing o mul iply sca e ed ligh when imaging
h ough u bid media.31,32 He e we do no in end o use he cohe -
ence ga e o mi iga e unwan ed sca e ing e ec s, bu we uncon en-
ionally shape he cohe ence ga e o ob ain sub-di ac ion limi ed
PSF.Fo ci cula ape u es,wecandesc ibe heCGFG(q)andpo(q)
using he Bessel unc ion o he i s kind as G(q)=2J1(μ)/(μ)
andpo(q)=2J1(ν)/(ν), whe e μ=2πKNAC∣q∣andν=2πKNAO∣q∣,
wi h NAC≤NAO.
To ob ain he sub-di ac ion limi ed esolu ion o QPI
a g{Γ[q;Δq=(0,0)]}, sys ems’s PSF h(q)=po(q)G∗(q)mus
ha e he cen al peak adius below he di ac ion limi . To main-
ain quan i a i e phase in o ma ion in he image, he sidelobes o
he PSF mus also be negligible. Nume ous s udies24,26,27,35 ha e
shown ha a supe oscilla o y ocal spo can be c ea ed by cohe en ly
illumina ing a phase o ampli ude mask composed o concen ic
annuli o di e en wid hs and complex ansmission. Supe oscilla-
ions a e hen o med by cons uc i e and des uc i e in e e ence
nea he ocal spo . As we use pa ially cohe en illumina ion in
ou mic oscope sys em, i is no possible o c ea e he supe oscil-
la o y ocal spo obse able in he ield’s in ensi y by in e e ence
as p oposed o cohe en ligh . Howe e , we can adop he p inci-
ples no mally applied o cohe en sys ems and shape he sys em’s
PSF, he p oduc o G∗(q−Δq)and po(q), by al e ing one o bo h
o hese unc ions. By modula ing he pupil unc ion POo(K )o
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APL Pho onics ARTICLE sci a ion.o g/jou nal/app
he objec -a m objec i e, we can a ec po(q), bu as Eq. (2) sug-
ges s, we ha e mo e op ions o G(q), because his unc ion can be
shapedbymodula ingoneo mo epupil unc ionsPCo(K ),PC (K )
and PO (K )o he condense s and he e e ence-a m objec i e,
espec i ely. I is also expe imen ally easie o modula e he con-
dense pupil planes. The e o e, we will ocus on shaping he CGF.
Howe e , simila esul s can be achie ed by shaping po(q), o bo h
a he same ime. Equa ion (2) desc ibing CGF o ma ion shows
ha G(q)can be shaped simila ly o cohe en imaging e en hough
he plane wa es exp(2πiK ⋅q)supe posed in Eq. (2) a e mu ually
incohe en . The P∗
Co(K )PC (K )PO (K )dic a es whe he hese
plane wa es a e cons uc i ely o des uc i ely supe posed. This
allows us o use app oaches designed o cohe en imaging e en
in a sys em ope a ing wi h pa ially cohe en ligh . The exp es-
sion in Eq. (2) is in ac an Ci e –Ze nike heo em,36 which
desc ibes he ela ionship be ween he mu ual cohe ence unc ion
(CGFin ou case)and he modula ion o hepupil plane o pa ially
cohe en b oad sou ce illumina ion. As we can con ol he con-
s uc i enesso heplanewa esupe posi ion, heo e ically,i should
be possible o c ea e obse able supe oscilla ions in pa ially cohe -
en sys ems. Howe e , no in he ield’s in ensi y bu in he mu ual
cohe ence o wo ields (in ou case, he CGF), hence he need o
he in e e ome ic sys em.
Fo demons a ion, we simula e he imaging and calcula e
he PSFs o h ee cases wi h di e en CGF shapes: i s , he
di ac ion-limi ed case, when a ull unmodula ed condense ape -
u e is assumed; second, he limi ing case o he supe oscilla ion,
when he ampli ude mask wi h na ow annulus is used and he CGF
is ep esen ed by he Bessel unc ion J0(2πKNAC∣q∣); and hi d,
he case wi h a supe oscilla o y CGF p oduced by h ee-zone phase
modula ion. Fo all h ee cases, we assume ha he po(q) unc-
ion is he Ai y pa e n o NAO=NAC=0.30, and his unc ion
is ep esen ed in Figs. 2(a)–2(c) by yellow dashed cu es. The CGF
G(q) o he di ac ion limi ed case is also he Ai y pa e n [see
he ed dashed cu e in Fig. 2(a)]. The CCHM PSF [ he p oduc
o po(q)and G(q)] is in Figs. 2(a)–2(c) depic ed by solid pu ple
cu es. The CGF desc ibed by J0(2πKNAC∣q∣), shown in Fig. 2(b),
can be p oduced in he Köhle a angemen by an annula inco-
he en sou ce wi h an in ini esimally na ow annulus and a adius
co esponding o he condense nume icalape u e NAC.Ou phase
modula ion app oach [Fig. 2(c)] o he c ea ion o supe oscilla o y
CGF is inspi ed by he esul s om Re . 35. We assume modula ion
o PC (K )by concen ic annuli wi h phase modula ion being ei he
0 o π. We ha e ound by a ew adjus men s and isual e alua ion
ha a supe oscilla o y CGF can be c ea ed by a h ee-zone annula
modula ionp oducedin he ollowingmanne : woci cleswi h adii
co esponding o 0.35NACand 0.72NACde ine he geome y o he
h ee zones, while he phase modula ion is 0 o he inne -mos and
ou e -mos zones, and he middle annulus has he phase shi o π
adians. The ull wid h a hal maximum o he cen al peak and
he i s ze o alue o he PSF de ine he sys em’s esol ing powe .
The h ee-zone phase modula ion and annula ampli ude modu-
la ion (annula sou ce) o he pupil unc ion PC (K )p oduce he
CGF wi h he sub-di ac ion limi ed cen al peak a he cos o
s onge sidelobes [see he ed dashed cu es in Figs. 2(b) and 2(c)].
I is impo an o no e ha e en hough hese unc ions hemsel es
a e supe oscilla o y, i he PSF wi h such s ong sidelobes is used
di ec ly o imaging, i p oduces unwan ed image a i ac s ha co -
up he imp o ed esol ing powe .37 Howe e , as we demons a e,
hepa iallycohe en sys emsp o idean elegan way o supp ess he
sidelobe e ec s.
The Ai y spo c ea ed by he objec i e in he objec a m has a
b oadcen alpeakwi hweaksidelobes.Asshownin hissec ion, he
PSF o he sys em ( he solid pu ple cu es in Fig. 2) is he p oduc o
heCGFand heAi y pa e n. In bo h annula sou ce [Fig. 2(b)] and
phase-modula ed [Fig. 2(b)] condense pupil cases, he CGF cen al
peak dic a es he sub-di ac ion limi ed p ope ies o he ocal spo ,
and sidelobes a e a enua ed by weak sidelobes o he Ai y pa -
e n dis ibu ion. The e o e, hese app oaches should p o ide sub-
di ac ion limi ed powe s and deli e single-sho supe - esolu ion
images.
We pe o med imaging simula ions compa ing h ee cases
co esponding o Figs. 2(a)–2(c) o e alua e he phase imaging pe -
o mance. We simula ed he phase esolu ion a ge imaging as
a cohe en con olu ion o i s complex ansmission unc ion by
he calcula ed PSFs, and he simula ed phase images a e shown in
Figs. 3(a)–3(c). The inse s in Figs. 3(b) and 3(c) show he po en ial
expe imen al design o he masks p oducing he simula ed modu-
la ion co esponding o Figs. 2(b) and 2(c). The smalles esol ed
elemen in he di ac ion-limi ed case is elemen 5 om g oup −2.
In he supe oscilla o y case [Fig. 3(c)], he smalles esol ed elemen
is numbe 1 om g oup −1, and o he annula pupil [Fig. 3(b)],
hiselemen canbe conside ed esol edwi h e ypoo con as .The
c oss-sec ionso he ea u eso elemen 6 omg oup −2inFig.3(d)
show ha his elemen is no esol ed in a di ac ion-limi ed image
bu well esol ed in bo h he annula pupil and supe oscilla o y
cases. Howe e , he supe oscilla o y PSF p oduces an image wi h
signi ican ly be e con as . The ea u e wid h o elemen 1 om
FIG. 2. The poin sp ead unc ion (PSF) is he p oduc o he cohe ence-ga ing unc ion (CGF) and he Ai y pa e n. (a) The s anda d imaging condi ion wi h a ull ape u e
condense . (b) An annula pupil condense p oduces sub-di ac ion limi ed CGF. (c) Phase modula ed pupil plane deli e s supe oscilla o y CGF and supe - esolu ion PSF.
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FIG. 3. Nume ical simula ions o phase esolu ion a ge imaging show ha he supe oscilla o y cohe ence-ga ing unc ion (CGF) c ea ed by phase modula ion p o ides
highe esol ing powe s, bu ampli ude modula ion is mo e obus in expe imen al si ua ions. (a) Di ac ion-limi ed quan i a i e phase image (QPI) o he phase esolu ion
a ge compu ed o he ull ape u e condense pupil. (b) QPI o he esolu ion a ge o ampli ude-modula ed condense pupil plane. (c) QPI o he esolu ion a ge
co esponding o he case o supe oscilla o y CGF c ea ed by phase modula ion. (d) and (e) P o iles o c oss-sec ions h ough elemen 6 o g oup −2 and elemen 1 o
g oup −1, espec i ely. Mo e de ails abou ou simula ions can be ound in he supplemen a y ma e ial.
g oup −1 is ∼20% lowe han he ea u e wid h o elemen 5 om
g oup −2. We can conclude ha he esolu ion imp o emen is
sligh ly less han 20% because he lines o elemen 1 om g oup
−1 ha e e y poo con as . As can be seen by compa ing Figs. 3(b)
and 3(c) and he c oss-sec ions in Fig. 3(e), he con as is be e in
he supe oscilla o y case. We expec ed he esolu ion o he case
wi h supe oscilla o y CGF o be be e because he annula pupil
p oduces he limi ing case CGF be ween supe oscilla o y and sub-
di ac ion limi ed ones. E en hough we achie e highe esol ing
powe wi h supe oscilla o y CGF, he o e all image quali y o he
supe oscilla o y case is lowe due o incomple e sidelobe a enua-
ion. I is impo an o no e ha , o simplici y o demons a ion,
we ha e no used any sophis ica ed me hods o op imize he con-
dense pupil unc ion. Gene a ion o supe oscilla o y ho spo s wi h
s a e-o - he-a pa ame e s usually employs i e a i e and compu-
a ionally expensi e p ocedu es such as pa icle swa m,27 gene ic
algo i hm,38 o phase e ie al39 op imiza ions.Weexpec oachie e
highe esolu ion imp o emen and be e phase image quali y by
employing one o hese me hods.
Addi ionally o spo size, he supe oscilla o y ocal spo design
always in ol es op imizing he a io o he cen al peak and he
sidelobe in ensi ies.29 Wi hou aking his in o accoun , p ac ical
applica ionso supe oscilla o y ocusing o imaginga eno possible
due o he poo signal- o-noise a io. A simila p inciple applies also
o op imizing he pa ame e s o he CGF. To econs uc QPI om
holog ams wi h easonable phase quali y, he holog am con as
mus be highe han he noise le els. When phase modula ion o he
pupil planes is used, he ampli ude o he CGF is edis ibu ed om
he cen al peak o he sidelobes due o he des uc i e in e e ence
o ligh om he objec and e e ence a m. The holog am con as is
p opo ional o he cen al peak ampli ude o he PSF h(q). The e-
o e, one mus conside he achie able holog am con as when
designing he supe oscilla o y CGF. We ha e disco e ed ha o he
combina ion o high-quali y phase and highes esolu ion imp o e-
men , i is impo an o op imize he whole p oduc o po(q)and
G(q), no only CGF G(q). Consequen ly, he objec i e unc ion o
an op imiza ion p ocedu e mus be de ined di e en ly han o a
s anda d in ensi y imaging sys em. One can easily deduce ha he
op imal solu ions ound o luo escence and con ocal mic oscopy
do no apply o he p oposed case.
Thec ea ion o he supe oscilla o yCGF equi esa e yp ecise
design o he phase modula ion o PC (K ). This is easily achie ed
in simula ions when unabe a ed pupils a e assumed. Howe e , we
ha e o accoun o abe a ions in eal expe imen al sys ems and
compensa e o hem while also p o iding he modula ion o CGF
shaping. Abe a ions can be pe cei ed in he con ex o he heo y
ou lined in his sec ion as modula ions o pupil unc ions PCo(K ),
PC (K ), and PO (K )in Eq. (2). As he c ea ion o supe oscilla ions
is e y suscep ible o e en sub le de ia ions om he designed phase
shi p o ided by he phase mask, he abe a ions p e en us om
using simple symme ic phase masks in eal sys ems. Due o he
di icul y o measu ing and compensa ing o abe a ion in ou sys-
em, we chose o u ilize he ampli ude mask in expe imen s ins ead
o he phase mask.
As Fig. 3 shows, he e ec o he modula ion by he mask can
beassessed indi ec ly om hesys em’simagingpe o mance.How-
e e , we can di ec ly measu e he shape o he CGF. When no spec-
imen is p esen in he objec a m, and we assume ha he objec i e
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lens in he objec a m has negligible abe a ions, hen we ge om
Eq. (1) he ollowing exp ession:
ΓB(Δq)=G∗(Δq).(3)
This equa ion shows ha he comple e MCF measu emen
ΓB(Δq)wi h no objec s p esen in bo h a ms p o ides us wi h
in o ma ion abou he CGF as a unc ion o Δq. The e o e, we will
use he measu emen desc ibed by Eq. (3) o di ec ly e alua e he
CGF shape c ea ed by he designed mask.
IV. EXPERIMENTAL RESULTS
We expe imen ally demons a e he easibili y o he p inciples
p oposed in Sec. III u ilizing he op ical se up wi h 10x/NAO=0.30
objec i es and he condense ape u e se o NAC=0.30. Ou ini-
ial e o s o ake ad an age o he phase modula ion p o ided by
simple phase masks simila o he one in he inse o Fig. 3(c) ha e
shown ha abe a ions in ou sys em p e en he CGF om being
shaped as designed. Howe e , ampli ude modula ion by an annula
mask [shown in Fig. 3(b) and he design pa ame e s in Sec. II] has
p o en ela i ely obus o he abe a ed pupils. The e o e, we used
i in he p esen ed expe imen s. We placed wo iden ical ampli ude
masks in o he e e ence and objec a ms o balance he ligh pow-
e s in he a ms in o de o achie e a be e con as o holog aphic
inges. The o al powe ac ion ha is ansmi ed o he specimen
h ough hemaskcanbecalcula edasa a ioo he anspa en mask
a ea o he ull ape u e a ea. As s a ed in Sec. II, he diame e o he
inne ci cle o he ampli ude annulus is 16.4 mm, and he e ec i e
condense ape u e diame e in he on ocal plane o he con-
dense is 17.3mm.The e o e, he a io o heligh ansmi ed o he
ligh inciden on he mask is ∼0.1. E en hough 90% ligh loss seems
signi ican , ou LED sou ce is powe ul enough o compensa e o
ha . In expe imen s wi h he ampli ude mask, we ope a ed he
sou ce a abou 10% o i s maximum powe , while he came a
exposu e imes did no exceed ens o milliseconds.
Fi s , we e alua e whe he he ampli ude modula ion p o ides
uswi haCGF esembling hedesignedshapeo J0(2πKNAC∣q∣).We
measu ed he comple e MCF o a case wi h [Fig. 4(b)] and wi hou
[Fig. 4(a)] he mask placed in he on ocal plane o condense s,
i.e., o annula and ull ape u e. We measu ed he comple e MCF
by acqui ing and econs uc ing a holog am o each e e ence a m
objec i e posi ion om a p ede ined g id. The g id o Δqposi ions
o each CGF measu emen was he same, and we used a 41 ×41
g id cen e ed a Δq=0 wi h a 0.3 μm spacing. We display in Fig. 4
he no malized modulus o he MCF o a FOV poin q=(0,0)μm.
Compa ing he measu ed CGF in Figs. 4(a) and 4(b) wi h he co e-
sponding simula ed CGF in Figs. 4(c) and 4(d), we see he e ec s o
he abe a ed pupils (mainly due o he o -axis holog aphic se up).
The measu emen wi h ull condense ape u es in Fig. 4(a) shows
clea signs o a p ima y coma abe a ion. We can conclude ha
he annula ape u e is no e y suscep ible o abe a ions, as he e
is a no able ag eemen be ween he measu ed [Fig. 4(b)] and sim-
ula ed [Fig. 4(d)] CGF p o iles. We ha e i ed he Ai y unc ion
o he measu ed CGF ampli ude, shown in Fig. 4(a), wi h signi i-
can lybe e sampling han hemeasu ed da a.Then,we de e mined
he ull wid h a hal maximum (FWHM) o he cen al peak o be
1.60 μm. Simila ly, we i ed he da a ob ained o he case wi h
FIG. 4. Compa ison o he measu ed and simula ed cohe ence-ga ing unc ion
(CGF) shows he e ec s o op ical abe a ions in he expe imen al se up on CGF.
(a) Measu ed CGF o ull condense ape u e. (b) Measu ed sub-di ac ion lim-
i ed CGF o annula condense ape u e. (c) Simula ed CGF o ull condense
ape u e. (d) Simula ed CGF o annula condense ape u e.
he annula ape u e, shown in Fig. 4(b), wi h he J0 unc ion
and de e mined he FWHM o be 1.08 μm. The measu emen in
Fig. 4(b) and he FWHM alues show ha he CGF c ea ed by he
ampli ude mask has a cen al peak na owe han he one o he
Ai y pa e n. The e o e, he CGF is, in his sense, sub-di ac ion
limi ed.
The comple e measu emen o he MCF, as shown in Fig. 4,
can be used o assess he op ical sys em abe a ions. This indica es
ha we could design and manu ac u e a phase mask ha would
simul aneously compensa e o abe a ions and p o ide he mod-
ula ion needed o supe oscilla o y CGF. Howe e , we decided o
pos pone hese e o s o ollow-up wo k as he cu en expe imen-
al se up limi s he p ac ical easibili y o his app oach. To ob ain
FIG. 5. Compa ison o (a) he di ac ion-limi ed image (ob ained wi h he ull ape -
u e condense ) and (b) he supe - esolu ion image (ob ained wi h he annula
mask) o he phase esolu ion a ge .
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FIG.6. Resul s o hep oposedme hod o imaging a emb yo ib oblas s. (a)FullFOVsupe - esolu ion QPI ob ained wi h sub-di ac ionlimi ed CGF.(b)Asec ion[ma ked
by he yellow dashed line in (a)] o he di ac ion-limi ed image. (c) A sec ion [ he same as in (b)] o he supe - esolu ion image. (d)A sec ion [ma ked by he ed dashed line
in (a)] o he di ac ion-limi ed image. (e) A sec ion [ he same as in (d)] o he supe - esolu ion image.
he supe oscilla o y esponse p o ided by a speci ically manu ac-
u ed asymme ic mask, one mus place i p ecisely in he co ec
posi ion. Se e al deg ees o eedom (pa ame e s) mus be se o
op imize he phase mask posi ion: axial, x-y ans e sal, and wo
angula posi ions.Fo his,wewouldneedau oma icalignmen wi h
a eedback loop. The ampli ude mask is easie o align as we can
pa ially see i s e ec in he in ensi y image o med by ligh om a
single mic oscope a m. Howe e , he phase mask e ec is no isible
in he in ensi y image. We can ake ad an age o he comple e MCF
measu emen desc ibed by Eq. (3) o assess he phase mask e ec
and i s posi ion. Un o una ely, his measu emen in he cu en
se up akes ens o minu es. The e o e, i is cu en ly unsui able o
implemen ing i in o a necessa y au oma ic alignmen p ocedu e
wi h a eedback loop.
We imaged a phase esolu ion a ge wi h bo h he ull and
annula condense ape u es o assess he imp o emen in he
esol ing powe . The ull (unmodula ed) ape u e phase image in
Fig. 5(a) ep esen s he di ac ion-limi ed image. The phase image
in Fig. 5(b) is ob ained using he ampli ude mask, which we e e
o as a supe - esolu ion image. The inse s in bo h images show
de ails o he smalles esol ed ea u es in each image. A isual
compa ison o Fig. 5(a) wi h Fig. 5(b) shows a clea esolu ion
imp o emen . The smalles esol able ea u es in he di ac ion-
limi ed image [Fig. 5(a)] ma ked by he numbe 20 a e 0.65 μm
wide. This means he esolu ion wi h he ull ape u e is ∼1.3 μm.
Whe eas he smalles esol able ea u es in he supe - esolu ion
image[Fig.5(b)]a ema kedby henumbe 22anda e0.53μmwide.
The imp o emen o he spa ial esolu ion o abou 1.06 μm is a
gain o ∼19%.
Nex , we show he pe o mance o ou me hod when used o
image complex specimens such as a emb yo ib oblas s in Fig. 6.
The p esen ed expe imen in ol ed LW13K2 cells om a cell line
o spon aneously ans o med a emb yo ib oblas s LW13 o he
inb ed s ain Lewis. Cells we e cul i a ed a 37○C in a humidi ied
incuba o wi h 3.5% CO2in s anda d Minimum Essen ial Medium
Eagle wi h Hanks’ sal s supplemen ed wi h 10% e al bo ine se um,
20 μM gen amicin, and 2 mM L-glu amine. Subsequen ly, he cells
we e ixed using 4% o maldehyde in phospha e-bu e ed saline o
20 min, hen washed and incuba ed in phospha e-bu e ed saline.
Again,we imaged he specimenwi h andwi hou heannula ampli-
ude masks in he on ocal condense planes. The QPI in Fig. 6(a)
expe iences supe - esolu ion h oughou he en i e FOV and can
be ob ained om a single holog am measu emen . Ha ing a la ge
FOV and su icien esolu ion o cell segmen a ion o e en obse -
a ion o in acellula p ocesses is c ucial when moni o ing, o
example, he mo ili y o li e cance cells.40 Compa ing he sec ions
o di ac ion-limi ed [Figs. 6(b) and 6(d)] and supe - esolu ion
[Figs. 6(c) and 6(e)] images, we see ha ou me hod p o ides he
imp o ed esolu ion equi ed in many applica ions in addi ion o
he la ge FOV.
V. CONCLUSION
In his pape , we ha e p esen ed a me hod o single-sho label-
ee supe - esolu ion QPI in holog aphic mic oscopes wi h pa ially
cohe en illumina ion. Ou solu ion o o e coming he di ac ion
limi is s aigh o wa d oimplemen becausei doesno equi e any
changes o he mic oscope’s op ical sys em. The p oposed me hod
elies on he in insic pa ially cohe en illumina ion p ope ies gi -
ing ise o he cohe ence-ga ing. We p opose ha by in oducing a
phase o ampli ude modula ion o he planes conjuga ed wi h he
ligh sou ce, e.g., he on ocal plane o he condense , we can gen-
e a e sub-di ac ion limi ed CGF. We demons a e o he i s ime
heo e ically and in nume ical simula ions a supe oscilla o y CGF
shaped by phase and ampli ude modula ion. Due o expe imen al
challenges, we chose o expe imen ally show he p oposed p inci-
ples using modula ion p o ided by an ampli ude mask, which has
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APL Pho onics ARTICLE sci a ion.o g/jou nal/app
p o en mo e obus o op ical abe a ions han phase masks. We
demons a ed almos 20% esol ing powe imp o emen in phase
imaging o he model specimen and complex objec s such as cance
cells. Howe e , he heo e ical spa ial esolu ion imp o emen is no
in p inciple limi ed, and we expec o ob ain signi ican ly o e 20%
esolu ion gain wi h mo e sophis ica ed modula ion echniques. Fo
example, a spa ial ligh modula o can be in oduced in o he op i-
calse up op o idesimul aneouscompensa iono pupilabe a ions
and he modula ion needed o c ea e he supe oscilla o y CGF.
We en ision ou me hod deli e ing an easily implemen able
supe - esolu ion QPI, pa icula ly sui able o high- h oughpu
biomedical applica ions. Fu he ex ension o he CGF shaping he-
o ybeyond he limi so hescala app oxima ionwillallow eaching
an unp eceden ed spa ial esolu ion o QPI. The possibili y o mon-
i o a la ge FOV in eal- ime wi h spa ial supe - esolu ion and e y
highquan i a i ein o ma ionquali ycansigni ican lyimpac cance
esea ch,14,41 asp e iouslyunseenin acellula p ocessescannowbe
obse ed. Fu he mo e, ou wo k sa is ies he need o ime-se ies
high-quali y da ase s equi ed o apidly de eloping au oma ed
analysis using a i icial in elligence.16,42
SUPPLEMENTARY MATERIAL
See he supplemen a y ma e ial o a de ailed de i a ion o he
equa ions in Sec. III, a de ini ion o he supe oscilla o y and supe -
esolu ion ocal spo ha is assumed h oughou his a icle, and
addi ional in o ma ion abou simula ions p oducing some o he
p esen ed da a.
ACKNOWLEDGMENTS
The wo k was suppo ed by he G an Agency o he Czech
Republic(G an No.21-01953S), he speci ic esea chg an s o B no
Uni e si y o Technology (G an Nos. FSI-S-20-6353 and FSI-S-
23-8389), and he MEYS CR (La ge RI P ojec No. LM2023050
Czech-BioImaging). We hank Ve onika J˚
uzo á o help in he
p epa a ion o biological samples.
AUTHOR DECLARATIONS
Con lic o In e es
R.C.isa co-au ho o pa en sco e ingQ-Phase (EA 018804 B1,
US8526003B2,JP5510676B2,CN102279555A,EP2378244B1,and
CZ302491) and a ecipien o ela ed oyal ies om Teligh .
Au ho Con ibu ions
Mi osla ˇ
Du iˇ
s: Concep ualiza ion (equal); Da a cu a ion (lead);
Fo mal analysis (lead); In es iga ion (lead); Me hodology (equal);
So wa e (lead); Visualiza ion (lead); W i ing – o iginal d a (lead);
W i ing – e iew & edi ing (equal). Pe Bouchal: Concep ual-
iza ion (equal); Funding acquisi ion (equal); In es iga ion (equal);
Me hodology (equal); Supe ision (equal); Valida ion (equal); W i -
ing – e iew & edi ing (equal). Radim Chmelík: Concep ualiza-
ion (equal); Fo mal analysis (equal); Funding acquisi ion (equal);
P ojec adminis a ion (equal); Resou ces (equal); Supe ision
(equal); W i ing – e iew & edi ing (equal).
DATA AVAILABILITY
The da a ha suppo he indings o his s udy a e a ailable
om he co esponding au ho upon easonable eques .
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